Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,31,30)(27,35,38,32)(28,33,37,39)(29,40,36,34);(1,2)(3,6)(4,7)(5,8)(11,17)(15,21)(18,23)(22,24)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,40)(34,39);(1,3)(2,5)(4,8)(6,7)(10,14)(15,21)(16,19)(22,24)(25,28,31,37)(26,33,30,39)(27,36,38,29)(32,40,35,34);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,31,36)(26,34,30,40)(27,37,38,28)(32,39,35,33);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,27)(26,35)(28,36)(29,37)(30,32)(31,38)(33,34)(39,40);(1,2)(3,5)(4,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,26)(27,35,38,32)(28,39,37,33)(29,40,36,34)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9575608);